Exponents and logarithms
You should know about exponents and logarithms from algebra.
In particular, if b is a positive number
and r is any real number,
then you should know about br;
if also b ≠ 1 and t is a positive number,
then you should know about logb t.
You should also know that these two equations are equivalent:
(The number b is called the base.)
Properties of exponents and logarithms
Fix a positive number b;
on the left-hand side below are properties of exponents
that hold for any real numbers r and s
and any positive number t.
On the right-hand side below,
suppose that b ≠ 1 (and still positive);
these properties of logarithms
hold for any real number r
and any positive numbers s and t:
|br > 0,
||blogb s =
|b0 = 1,
||logb 1 = 0;|
|br + s =
||logb (st) =
logb s +
||logb (1/t) =
|br − s =
||logb (s/t) =
logb s −
|b1 = b,
||logb b = 1;|
||logb (tr) =
r logb t;|
||logb (t√s) =
logb s ÷
In addition, there are these change-of-base rules:
So in a way, you only need to know about exponents and logarithms
for one specific base e.
In calculus, it's convenient to choose
the special constant e ≈ 2.7;
we write ln t for loge t
and (sometimes) exp r for er.
- If b and e are positive, e ≠ 1,
and r is any real number,
then br =
er loge b;
- if b, e, and t are positive,
b ≠ 1, and e ≠ 1,
then logb t =
loge t ÷
So you should know at least the algebraic rules
for exponents and logarithms with base e,
and how to convert any other expression into base e:
(Here, b and t are positive,
and b ≠ 1 in the last rule.)
- br =
exp (r ln b);
- logb t =
ln t ÷ ln b.
Rules for differentiation
There are two basic ways
to go about differentiating expressions with exponents and logarithms.
One way is to turn everything into
a more complicated expression where the only base is e
and learn rules for those expressions only:
These are the simplest rules to learn,
but before you can use them,
you may have to do some algebra to change the base.
- d(exp u) =
exp u du;
- d(ln u) = du ÷ u.
The other way is to learn rules
for any exponential or logarithmic expression:
These are more complicated, but you can always use them right away.
- d(vu) =
vu−1 (v ln v du + u dv);
- d(logv u) =
(v ln v du − u ln u dv) ÷
(uv ln2 v).
It may also be handy to learn
the special cases of the above rules
when one part of the expression is constant
(which can then be simplified a bit):
The first of these four is our old friend the Power Rule;
it's the only one out of all of the rules above
that makes sense when the base v is negative
(in which case k must be a rational number
whose denominator, in lowest terms, is odd).
- d(vk) =
- d(logv k) =
−ln k dv ÷
(v ln2 v);
- d(ku) =
ku ln k du,
- d(logk u) =
du ÷ (u ln k).
Rules for limits
You can take limits of exponential and logarithmic expressions
by simply plugging in the limiting value, as usual,
so long as the base of every such expression is positive.
But even when the original expressions are positive,
it's possible that the limit of the base is zero, so watch out!
In other cases, I advise you to change the base to e
and use these rules to deal with infinities:
It's possible to learn more complicated rules for arbitrary bases,
but there are far too many cases than I would want to handle.
- exp ∞ = ∞,
- ln ∞ = ∞;
- exp (−∞) = 0+,
- ln (0+) = −∞.
To handle indeterminate forms, we can use L'Hôpital's Rule.
In the case of a logarithm,
L'Hôpital will apply directly (once you switch to base e)
if it's indeterminate;
in the case of an exponential expression,
use this version of the change-of-base rule:
then you can apply L'Hôpital to the exponent.
Note especially the indeterminate form 00.
(Every other indeterminate form of exponentiation or logarithms
involves infinity somewhere.)
Since the base 0 is not positive,
you cannot simply evaluate this to 1.
- br =
exp (r ln b) =
exp (ln b ÷ (1/r));
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This web page was written between 2011 and 2015 by Toby Bartels,
last edited on 2015 August 15.
Toby reserves no legal rights to it.
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